On a normed version of a Rogers–Shephard type problem

Abstract

A translation body of a convex body is the convex hull of two of its translates intersecting each other. In the 1950s, Rogers and Shephard found the extremal values, over the family of n-dimensional convex bodies, of the maximal volume of the translation bodies of a given convex body. In our paper, we introduce a normed version of this problem, and for the planar case, determine the corresponding quantities for the four types of volumes regularly used in the literature: Busemann, Holmes-Thompson, and Gromov's mass and mass*. We examine the problem also for higher dimensions, and for centrally symmetric convex bodies.1991 Mathematics Subject Classification. 52A38, 52A21, 52A40.

The Isoperimetric Quotient of a Convex Body Decreases Monotonically Under the Eikonal Abrasion Model

“…decreases monotonically for any initial convex body K(0). Here V (·) denotes ddimensional Euclidean volume, and we remark that every finite dimensional normed space can be equipped by a Haar measure and that this measure is unique up to multiplication of the standard Lebesgue measure V (·) by a scalar [22,27].…”